# nmfsc: Non-negative Sparse Matrix Factorization In fabia: FABIA: Factor Analysis for Bicluster Acquisition

## Description

nmfsc: R implementation of nmfsc.

## Usage

 1 nmfsc(X,p=5,cyc=100,sL=0.6,sZ=0.6) 

## Arguments

 X the data matrix. p number of hidden factors = number of biclusters; default = 5. cyc maximal number of iterations; default = 100. sL sparseness loadings; default = 0.6. sZ sparseness factors; default = 0.6.

## Details

Non-negative Matrix Factorization represents positive matrix X by positive matrices L and Z that are sparse.

Objective for reconstruction is Euclidean distance and sparseness constraints.

Essentially the model is the sum of outer products of vectors:

X = ∑_{i=1}^{p} λ_i z_i^T

where the number of summands p is the number of biclusters. The matrix factorization is

X = L Z

Here λ_i are from R^n, z_i from R^l, L from R^{n \times p}, Z from R^{p \times l}, and X from R^{n \times l}.

If the nonzero components of the sparse vectors are grouped together then the outer product results in a matrix with a nonzero block and zeros elsewhere.

The model selection is performed by a constraint optimization according to Hoyer, 2004. The Euclidean distance (the Frobenius norm) is minimized subject to sparseness and non-negativity constraints.

Model selection is done by gradient descent on the Euclidean objective and thereafter projection of single vectors of L and single vectors of Z to fulfill the sparseness and non-negativity constraints.

The projection minimize the Euclidean distance to the original vector given an l_1-norm and an l_2-norm and enforcing non-negativity.

The projection is a convex quadratic problem which is solved iteratively where at each iteration at least one component is set to zero. Instead of the l_1-norm a sparseness measurement is used which relates the l_1-norm to the l_2-norm.

The code is implemented in R.

## Value

 object of the class Factorization. Containing LZ (estimated noise free data L Z), L (loadings L), Z (factors Z), U (noise X-LZ), X (data X).

Sepp Hochreiter

## References

Patrik O. Hoyer, ‘Non-negative Matrix Factorization with Sparseness Constraints’, Journal of Machine Learning Research 5:1457-1469, 2004.

D. D. Lee and H. S. Seung, ‘Algorithms for non-negative matrix factorization’, In Advances in Neural Information Processing Systems 13, 556-562, 2001.

fabia, fabias, fabiap, fabi, fabiasp, mfsc, nmfdiv, nmfeu, nmfsc, extractPlot, extractBic, plotBicluster, Factorization, projFuncPos, projFunc, estimateMode, makeFabiaData, makeFabiaDataBlocks, makeFabiaDataPos, makeFabiaDataBlocksPos, matrixImagePlot, fabiaDemo, fabiaVersion
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 #--------------- # TEST #--------------- dat <- makeFabiaDataBlocks(n = 100,l= 50,p = 3,f1 = 5,f2 = 5, of1 = 5,of2 = 10,sd_noise = 3.0,sd_z_noise = 0.2,mean_z = 2.0, sd_z = 1.0,sd_l_noise = 0.2,mean_l = 3.0,sd_l = 1.0) X <- dat[[1]] Y <- dat[[2]] X <- abs(X) resEx <- nmfsc(X,3,30,0.6,0.6) ## Not run: #--------------- # DEMO #--------------- dat <- makeFabiaDataBlocks(n = 1000,l= 100,p = 10,f1 = 5,f2 = 5, of1 = 5,of2 = 10,sd_noise = 3.0,sd_z_noise = 0.2,mean_z = 2.0, sd_z = 1.0,sd_l_noise = 0.2,mean_l = 3.0,sd_l = 1.0) X <- dat[[1]] Y <- dat[[2]] X <- abs(X) resToy <- nmfsc(X,13,100,0.6,0.6) extractPlot(resToy,ti="NMFSC",Y=Y) ## End(Not run)